Reconstruction of Order Statistics in Exponential Distribution

Transcription

1 JIRSS 200) Vol. 9, No., pp 2-40 Reconstruction of Order Statistics in Exponential Distribution M. Razmkhah, B. Khatib, Jafar Ahmadi Department of Statistics, Ordered and Spatial Data Center of Excellence, Ferdowsi University of Mashhad, Iran. razmkhah Abstract. In this article, a new censoring scheme is considered, namely, a middle part of a random sample is censored. A treatment for reconstructing the missing order statistics is investigated. The proposed procedure is studied in detail under exponential distribution which is widely used as a constant failure model in reliability. Different approaches are used to obtain point and interval reconstructors and then they are compared. A numerical example is presented for illustrating all the proposed inferential procedures. Eventually, we present some remarks including how the results of the paper can be used when the parameters of the exponential distribution are unknown. Key words and phrases: Beta distribution, highest conditional density, Markov property, pivotal quantity.

2 22 Razmkhah, et al. Introduction There are some situations in life testing and reliability experiments in which a middle part of the sample or subjects) are lost or removed from the experiment. For example, in a life testing experiment, suppose n items are placed on test simultaneously. The first few observations may be observed at the beginning of the experiment, then some other data points may be censored due to negligence or problems, while the last few observations are recorded. In such situations, the experimenter may not obtain complete information on failure times for all experimental units. What motivated us to write the paper is that How can one reconstruct the missing observations?. This scheme is the complementary to the idea of double censoring in which the middle part of the sample is actually stored. The doubly censored data model has been studied by several authors, see for example, Fernández 2004) and Sun et al. 2008). Meanwhile, the proposed scheme can be considered as a special case of middle censoring model which was first introduced by Jammalamadaka and Mangalam 2003) for which all random intervals are the same. The later plan is studied by Jammalamadaka and Iyer 2004) and Mangalam et al. 2008) in nonparametric set up and Iyer et al. 2008) in the parametric set up. Now, suppose n independent and identical units are placed on a life test with corresponding lifetimes X,..., X n with probability density function pdf) f and cumulative distribution function cdf) F. Denote the ith order statistic of the sample X,..., X n by Y i. Assume that some order statistics are lost, that is we only observed the data set Y = {Y,..., Y r, Y s,..., Y n }, where 0 r < s n +, for convenience of notation, we let Y 0 = 0 and Y n+ =. If r = 0, we indeed observe {Y s,..., Y n } which coincides with the left censoring model and in the case s = n +, the data set {Y,..., Y r } is observed where is the same as Type II censored data. In life testing and survival analysis, several researches have been done based on left and right censored data. The main goal of this paper is to reconstruct the value of Y l for r < l < s based on observed ordered data Y while the underlying distribution is exponential. A random variable X is said to have a two-parameter exponential distribution, which we shall write X Expµ, σ), if its cdf is F x) = e x µ σ, x µ, σ > 0, )

3 Reconstruction of Order Statistics in Exponential Distribution 23 where µ and σ are the location and scale parameters, respectively. It is well known that the exponential distribution is the simplest and most important distribution in reliability studies, and is applied in a wide variety of statistical procedures, especially in life testing problems. See Balakrishnan and Basu 995) for some researchs based on this distribution. The rest of this paper is as follows: In Section 2, some preliminaries are presented. Three point reconstructors for the censored data points of the exponential distribution are given in Section 3. Section 4 is focused on the interval reconstruction. In order to illustrate the proposed scheme, we present a numerical example in Section 5. At the end, we present some remarks including how the results of the paper can be used when the parameters of the exponential distribution are unknown. 2 Some Preliminaries Here, we present some well known properties of order statistics which will be used to obtain the new results in the next sections. Let X,..., X n be independent and identically distributed iid) continuous random variables with cdf F θ x) and pdf f θ x). In order to reconstruct the lth r < l < s) order statistic based on the data set Y, we propose various methods and obtain some reconstructors of Y l. First of all, notice that by Markove property of order statistics see, David and Nagaraja, 2003), the conditional pdf of Y l given Y is equivalent to that of Y l given Y r and Y s. Hence, f Yl Yy l y) = f Yl Y r,y s y l y r, y s ) F yl ) F y r ) ) l r F ys ) F y l ) ) s l = Bl r, s l) F y s ) F y r ) ) s r fy l ), y r < y l < y s, 2) where y = y,..., y r, y s,..., y n ) is the observed value of Y and Ba, b) is the complete beta function. From Eq. 2), it is deduced that the distribution of Y l given Y is just the distribution of the l r)th order statistic in a sample of size fx) s r drawn from a population with pdf F y s ) F y r ), y r < x < y s i.e., from the parent distribution truncated in the tails at y r and y s );

4 24 Razmkhah, et al. Also for 0 r < l < s n +, we have F Y l ) F Y r ) Y Betal r, s l), 3) F Y s ) F Y r ) where Betaa, b) denotes a beta distribution with parameters a > 0 and b > 0. It is obvious that the conditional distribution of F Y l ) F Y r) F Y s ) F Y r ) given Y is identical to the unconditional distribution of the l r)th order statistic in a sample of size s r from the standard uniform distribution. Now, suppose that Z r,s Betas r, n s+), then for 0 < m < we have E [ log mz r,s ) ] = Bs r, n s + )m n r s r n s ) ) s r n s ) i i j i=0 j=0 m )n s j i + j + ) 2 { m) i+j+ i + j + ) log m) ) } = φ r, s, m), say. 4) Also E [ log 2 mz r,s ) ] 2 = Bs r, n s + )m n r s r n s ) ) s r n s ) i m ) n s j i j i + j + ) 3 i=0 j=0 { m) i+j+ i + j + ) log m) )} +0.5i + j + ) 2 log 2 m) = φ 2 r, s, m), say. 5) 3 Point Reconstruction To reconstruct the missing order statistics from a middle part of a random sample based on the data set Y, in this section we propose three approaches and then compare them.

5 Reconstruction of Order Statistics in Exponential Distribution Convex Combination Reconstructor If the data set Y = {Y,..., Y r, Y s,..., Y n } is observed. Using the structure dependence properties of order statistics, it is logical to assume that a convex combination CC) of Y r and Y s may be considered as a reconstructor for Y l, denoted by Ŷl,CC, i.e., Ŷ l,cc = wy r + w)y s, r < l < s, 0 < w <. The main question arises: how can one choose w? It is reasonable to select the optimal value of w which can be determined by minimizing the mean squared error MSE) of Ŷl,CC. Notice that MSEŶl,CC) = EŶl,CC Y l ) 2 = E [ w)w l,s ww r,l ] 2 = w 2 EW 2 r,l ) + w)2 EW 2 l,s ) 2w w)ew r,l )EW l,s ), 6) where W r,l = Y l Y r and the last equality deduces from the independency of W r,l and W l,s in an exponential model spacing of order statistics). We recall that if X,..., X n are iid random variables from Expµ, σ), then W r,l can be expressed as W r,l = l i=r+ Z i n i +, 7) where Z i s are iid random variables from Exp0, σ). Using 7), we readily have and σ EW r,l ) = σ 2 EW 2 r,l ) = l i=r+ l i=r+ n i + = φ 3r, l), say 8) n i + ) 2 + φ2 3r, l) = φ 4 r, l), say. 9) Substituting 8) and 9) into 6), we have σ 2 MSEŶl,CC) = w 2 φ 4 r, l) + w) 2 φ 4 l, s) 2w w)φ 3 r, l)φ 3 l, s). 0)

6 26 Razmkhah, et al. The optimal value of w may be obtained by minimizing 0) with respect to w which is given by w opt = φ 4 l, s) + φ 3 r, l)φ 3 l, s) φ 4 r, l) + φ 4 l, s) + 2φ 3 r, l)φ 3 l, s), ) where φ 3 r, l) and φ 4 r, l) are defined in 8) and 9), respectively. For given n, r, s and l, one can easily find the values of w opt from ). We derive the values of σ 2 MSEŶl,CC) for n = 0 and some selected values of r, s and l with corresponding w opt, the results are presented in Table. 3.2 Conditional Median Reconstructor In the context of prediction, the conditional median prediction approach has been studied by several authors, see for example Raqab and Nagaraja 995), Raqab et al. 2007) and Ahmadi et al. 2009). Therefore, intuitively, the median of the conditional density of Y l given Y can be considered as a reconstructor of Y l. So we say that Ŷ l,cm is a conditional median CM) reconstructor of Y l, if P {Y l Ŷ l,cm Y} = P {Y l Ŷl,CM Y}. Using 3), we have Ŷ l,cm = F {F Y r ) + medv r,l,s )[F Y s ) F Y r )]}, 2) where V r,l,s Betal r, s l) and medx) stands for the median of X. By 2), the CM reconstructor of the lth order statistic in the exponential distribution ) is given by Ŷ l,cm = Y r σ log medv r,l,s ) e Wr,s/σ)). It is obvious that the pdf of W r,l is f Wr,l w) = e n l+)w/σ e w/σ ) l r, w > 0. 3) σbl r, n l + ) Using 3), we obtain the MSE of Ŷl,CM which is given by σ 2 MSEŶl,CM) = φ 4 r, l) + φ 2 r, s, medvr,l,s ) ) +2φ 5 r, l, s, medvr,l,s ) ),

7 Reconstruction of Order Statistics in Exponential Distribution 27 where φ 2 r, l) and φ 4 r, s, ) are defined in 5) and 9), respectively, and where φ 5 r, l, s, m) n! = r )!l r )!s l )!n s)! = 0 z y log x y ) log m z x )) x) r x y) l r y z) s l z n s dxdydz n r)! l r )!s l )!n s)! l r i=0 s l j=0 { i + j + ) 2 i + j + φ 6 r, l, s, m, i) = ) l r s l i j ) ) s l +i j ) φ 6 r, l, s, m, j ) φ 6 r, l, s, m, i) φ 6 r, l, s, m, i) log m + φ 7 r, l, s, m, i) )},4) n l+i ) n l + i m ) n l+i k m n l+i+ k k + ) 2 k=0 { m) k+ [ k + ) log m) ] } and φ 7 r, l, s, m, i) = n l+i m n l+i+ k=0 log m) 0 n l + i k ) m ) n l+i k x e k+)x logm + e x )dx. The numerical values of σ 2 MSEŶl,CM) are presented in Table for n = 0 and some selected values of r, s and l.

8 28 Razmkhah, et al. 3.3 Unbiased Conditional Reconstructor It is logical to consider E[Y l Y] as a reconstructor of Y l having observed Y.Whis this in mind, using 3), we consider Ŷ l,uc = F F Y r ) + l r ) s r [F Y s) F Y r )], 5) as an unbiased conditional UC) reconstructor of Y l. From 5), the UC reconstructor of the lth order statistic in Expµ, σ) is given by Ŷ l,uc = Y r σ log l r e W )) r,s/σ. s r Using 3), we obtain MSEŶl,UC) which is stated as σ 2 l r ) l r ) MSEŶl,UC) = φ 4 r, l) + φ 2 r, s, + 2φ5 r, l, s,, s r s r where φ 2 r, s, ), φ 4 r, l) and φ 5 r, l, s, ) are defined in 5), 9) and 4), respectively. We computed the numerical values of σ 2 MSEŶl,UC) for n = 0 and some choices of r, s and l, which are calculated in 4 decimal places using the package R, the results are presented in Table. Table. The numerical values of σ 2 MSEŶl,CC), σ 2 MSEŶl,CM ) and σ 2 MSEŶl,UC) for n = 0. r = 3 r = 4 l l s a b c

9 Reconstruction of Order Statistics in Exponential Distribution 29 r = 5 l s a, b and c indicate for σ 2 MSEŶl,CC), σ 2 MSEŶl,CM ) and σ 2 MSEŶl,UC), respectively. From Table, it is observed that. For l = r + < s, MSEŶl,CC) < MSEŶl,CM), but for r + < l < s it is usually vice versa. 2. It is usually observed that MSEŶl,UC) < MSEŶl,CC). 3. For fixed r and s, the MSEs of Ŷl,CC, Ŷl,CM and Ŷl,UC increase as l increases. 4. For fixed r and l, the MSEs of Ŷl,CC, Ŷl,CM and Ŷl,UC increase as s increases. 5. For fixed s and l, the MSEs of Ŷl,CC, Ŷl,CM and Ŷl,UC decrease as r increases. 6. For fixed r and s, the MSE of Ŷl,UC is usually less than, equal to and greater than that of Ŷl,CM when l is less than, equal to and greater than r+s 2, respectively. Specially, when s r is even, MSEŶ r+s,cm) = MSEŶ r+s,uc) Reconstruction Interval In this section, we are interested in finding the reconstruction intervals for the lth r < l < s) order statistic in terms of observed data set Y. Two methods are proposed and then compared in view of shortest width criterion.

10 30 Razmkhah, et al. 4. Conditional Reconstruction Interval We say that the interval [L, U] is the exact 00 α α 2 )% conditional reconstruction interval CRI) for Y l given Y if P Y l L Y} = α and P Y l U Y} = α 2. Using 3), it can be shown that and L CRI = F {F Y r ) + Bl r, s l, α )[F Y s ) F Y r )]}, 6) U CRI = F {F Y r ) + Bl r, s l, α 2 )[F Y s ) F Y r )]}, 7) where Bl r, s l, γ) is the 00γ% lower percentile of the Betal r, s l) distribution, i.e., P [ V r,l,s Bl r, s l, γ) ] = γ. For two-parameter exponential distribution, using the Eqs. 6) and 7), we obtain the lower and upper bounds of the exact 00 α α 2 )% CRI, respectively, as L CRI = Y r σ log Bl r, s l, α ) e W r,s/σ )), 8) and U CRI = Y r σ log Bl r, s l, α 2 ) e Wr,s/σ)). 9) Hence, the expected width of the CRI, EW CRI ) = EU CRI L CRI ), is { Bl r, s l, α ) e W r,s/σ ) )} EW CRI ) = σe log Bl r, s l, α 2 ) e ) W r,s/σ = σ { φ r, s, Bl r, s l, α ) ) φ r, s, Bl r, s l, α2 ) )}, 20) where φ r, s, ) is defined in 4). Similarly, E [ WCRI 2 ] = σ 2{ φ 2 r, s, Bl r, s l, α ) ) φ 2 r, s, Bl r, s l, α2 ) ) 2φ r, s, Bl r, s l, α ) ) φ r, s, Bl r, s l, α2 ) )}, 2) where φ 2 r, s, ) is defined in 5). Using 20) and 2), we can obtain variance of the width of the CRI. Table 2 shows the numerical values of σ EW CRI ) and σ 2 VarW CRI ) for α = α 2 = 0., n = 0 and some choices of r, s and l.

11 Reconstruction of Order Statistics in Exponential Distribution 3 Table 2. The numerical values of σ EW CRI ) and σ 2 VarW CRI ) for the 80% CRI, when n = 0. r = 3 r = 4 l l s a b r = 5 l s a and b indicate for the value of σ EW CRI ) and σ 2 VarW CRI ), respectively. From Table 2, it is observed that. For fixed r and s, EW CRI ) and VarW CRI ) increase as l increases. 2. For fixed r and l, EW CRI ) increases and VarW CRI ) decreases as s increases. 3. For fixed l and s, EW CRI ) and VarW CRI ) decrease as r increases. 4.2 Highest Conditional Density Reconstruction Interval Similar to the idea of highest conditional density prediction interval, the highest conditional density reconstruction interval HCDRI) may be considered such that the conditional pdf of Y l given Y for every point inside the interval is greater than that for every point outside of it. If the conditional pdf of Y l given Y is unimodal, it is sufficient to

12 32 Razmkhah, et al. derive an optimal 00 α)% reconstruction interval [L, U ], such that { P {L < Y l < U Y = y} = α, 22) f Yl YL y) = f Yl YU y). According to the Eqs. in 22) and using 3), the lower and upper bounds of the HCDRI can be determined such that the following equations are held B l r, s l, F U ) F Y r) F Y s) F Y r) F L ) F Y r ) F U ) F Y r ) ) l r ) ) B l r, s l, F L ) F Y r) F Y s) F Y r) = α, F Y s ) F L ) F Y s ) F U ) ) s l = fu ) fl ). Taking v = F L ) F Y r) F Y s) F Y and v r) 2 = F U ) F Y r) F Y s) F Y r), the HCDRI with coefficient α for Y l given Y is F [ F Y r )+v F Ys ) F Y r ) )], F [ F Y r )+v 2 F Ys ) F Y r ) )]), 23) where v and v 2 can be determined by solving the following two equations Bl r, s l, v 2 ) Bl r, s l, v ) = α, [ )]) v v 2 ) l r v v 2 ) s l = f F F Y r )+v 2 F Y s ) F Y r ) )]). f F [F Y r )+v F Y s ) F Y r ) Now, let X,..., X n be iid random variables with exponential distribution ). Using 23), the HCDRI with coefficient α for Y l given Y can be found as follows [ Y r σ log v e W r,s /σ )), Y r σ log v 2 e W r,s /σ ))], 24) such that v and v 2 are the solutions of the following two equations Bl r, s l, v 2 ) Bl r, s l, v ) = α, v v 2 ) l r v v 2 ) s l = v 2 e ) W r,s/σ ) 25), v e W r,s/σ

13 Reconstruction of Order Statistics in Exponential Distribution 33 provided > v 2 > v > 0, otherwise the HCDRI does not exist. Here we consider three special cases as follows. Case. Suppose that only one order statistic in a sample of size n is missed and we are looking to reconstruct it. That is, we consider the reconstruction of Y l on the basis of the data set Y such that r = l and s = l +. In this case, the conditional pdf of Y l given Y is f Yl Yy l y) = e y l/σ { e y r/σ e ys/σ}, yr < y l < y s, σ which is a decreasing function of y l. Hence, there is not any two-sided HCDRIs for Y l given Y. But, one can obtain a one-sided one with coefficient α as follows [ Y r, Y r σ log α) e Wr,s/σ))]. 26) The expected width and variance of the one-sided HCDRI 26) are given by σφ r, s, α) and σ 2 {φ 2 r, s, α) φ r, s, α) 2 }, respectively, where φ r, s, ) and φ 2 r, s, ) are defined in 4) and 5), respectively. Case 2. Suppose that only two order statistics in a sample of size n are missed and we are looking to reconstruct the smallest one. That is, we consider the reconstruction of Y l based on the data set Y such that r = l and s = l + 2. Notice that in this case, the conditional pdf of Y l given Y is a decreasing function of y l on the interval y r, y s ). Similar to the Case, a one-sided HCDRI with coefficient α can be found which is [ Y r, Y r σ log α) e Wr,s/σ))]. 27) The expected width and variance of the one-sided HCDRI 27) are σφ r, s, α) and σ 2 {φ 2 r, s, α) [φ r, s, α)] 2 }, respectively. Case 3. Suppose that the assumption of Case 2 holds and we are attempting to reconstruct the largest one. That is, we consider the reconstruction of Y l in terms of the data set Y such that r = l 2 and s = l +. In this case, the conditional pdf of Y l given Y is f Yl Yy l y) = 2 e yr/σ e y l/σ e y r/σ e ys/σ) 2 e y l/σ σ, y r < y l < y s. 28)

14 34 Razmkhah, et al. It can be shown that the conditional pdf in 28) is an increasing function of y l on the interval y r, y s ) whenever W r,s = w r,s < σ log 2, otherwise it is a unimodal pdf. Therefore, we consider the following two cases. i) W r,s = w r,s < σ log 2. In this case, a one-sided HCDRI for Y l given Y with coefficient α may be determined as follows [ Y r σ log α e W r,s/σ )), Y s ]. 29) The expected width of the one-sided HCDRI 29), denoted by EW ), is EW ) = σ { φ r, s, α) + φ 3 r, s) }, 30) where φ r, s, ) and φ 3 r, s) are defined in 4) and 8), respectively. Also, EW 2 ) = σ 2{ φ 4 r, s) + φ 2 r, s, α) + 2φ 8 r, s, α) }, 3) where φ 2 r, s, ) and φ 4 r, s) are defined in 5) and 9), respectively, and φ 8 r, s, m) = Bs r, n s + )m n r s r i=0 n s ) s r n s i j j=0 ) ) i { log m m ) n s j i + j + ) 2 )] [ m) i+j+ i + j + ) log m) log m) 0 } xe i+j+)x logm + e x )dx. 32) ii) W r,s = w r,s > σ log 2. In this case a two-sided HCDRI for Y l given Y can be found in the form of 24). By 25), we find v and v 2 by solving the following two equations v 2 2 v2 = α, v v e W r,s /σ )) = v 2 v 2 e W r,s /σ )). 33)

15 Reconstruction of Order Statistics in Exponential Distribution 35 Then, ) v = α) e W 2 r,s/σ e ), W r,s/σ 2 v 2 = + α) e Wr,s/σ) 2 2 e Wr,s/σ). 34) It is obvious that v < v 2. By investigating the conditions in 25), i.e., 0 < v, v 2 <,we deduce that an HCDRI exists if and only if α < 2e Wr,s/σ) e Wr,s/σ) 2, provided W r,s = w r,s > σ log 2. Then, by substituting 34) into 24), the HCDRI is given by [ Y r σ log + α) e W r,s /σ ) 2 2 α) e W r,s /σ ) 2 )] Y r σ log. 35) 2 The expected width of the two-sided HCDRI 35), denoted by EW 2 ), is where Also, EW 2 ) = σ { φ 9 r, s, α ) φ 9 r, s, α) }, 36) φ 9 r, s, m) = ), Bs r, n s + ) n s n s ) ) i log mx 2 )x s r+i dx. i i=0 0 EW 2 2 ) = σ 2{ φ 0 r, s, α ) + φ 0 r, s, α) 2φ r, s, α) },37) where φ 0 r, s, m) = Bs r, n s + ) n s n s ) ) i x s r+i log 2 mx 2 )dx i i=0 0

16 36 Razmkhah, et al. and φ r, s, m) = Bs r, n s + ) n s ) n s ) i i i=0 0 x s r+i log mx 2 ) log + mx 2 )dx. Now, we can determine the expected value and the variance of the width of the HCDRI for the Case 3. Denote the width of the HCDRI in this case by W HCDRI, then we have, for k, EW k HCDRI) = pn, r, s)ew k ) + pn, r, s) ) EW k 2 ), 38) where from the relation between beta and binomial distributions, we have n ) n r pn, r, s) = P W r,s < σ log 2) = 2 r n. i r Using 30), 3), 36), 37) and 38), the exact values of σ EW HCDRI ) and σ 2 VarW HCDRI ) can be obtained. The results are presented in Table 3 for 80% HCDRIs when n = 0. Table 3. The numerical values of σ EW HCDRI ) and σ 2 VarW HCDRI ) for 80% HCDRIs when n = 0. r = 3 r = 4 r = 5 l l l s a b i=s a and b indicate for σ EW HCDRI) and σ 2 VarW HCDRI), respectively. Remark 4.. Comparing Tables 2 and 3, it is observed that for fixed r, l and s, the expected width of the HCDRI is less than that of the corresponding CRI.

17 Reconstruction of Order Statistics in Exponential Distribution 37 5 Numerical Example To illustrate the performance of the proposed reconstructors in Sections 3 and 4, we use the data presented in Table 4 denoting the ordered observations of a random sample of size 0 generated from Exp2, 5). Table 4. Ordered observations of a random sample of size 0 generated from Exp2, 5). i Y i Suppose that we only observe {Y,, Y 4, Y 7,, Y 0 }. Then, for l = 5, 6, we reconstruct Y l using different mentioned approaches in the previous sections. As shown in subsection 4.4, by 27) we find a one-sided HCDRI for Y 5. Also, we can obtain a two-sided HCDRIs 2e for Y 6 with reconstruction coefficient at most w 4,7 e /σ w /σ) 2 = ,7 provided W 4,7 = w 4,7 > 5 log 2 = 3.466, for which σ = 5 is known and w 4,7 = y 7 y 4 = In the case that the scale parameter σ is unknown, one can plug in the common estimator of σ, see the next section, so using Eq. 43), we can obtain MLEσ) In this case an HCDRI for Y 6 can be found with coefficient at most Table 5 contains the values of different reconstructors of Y l The reconstruction coefficient α = 0.80 is considered). Table 5. The values of different reconstructors of Y l. l Y l exact value) Ŷ l,cc Ŷ l,cm Ŷ l,uc CRIY l ) HCDRIY l ) a , ) 4.65, 6.729) b , ) 4.65, 6.802) , ) , ) , ) 6.92, ) a, b and indicate for σ is known, σ is unknown and one-sided HCDRI, respectively. From Table 5, it is observed that. Among the point reconstructors of Y 5, Ŷ5,CC is the closest to Y Among the point reconstructors of Y 6, Ŷ6,UC is the closest to Y The width of HCDRIs for Y 5 and Y 6 are less than those of CRIs in the both cases that σ is known or unknown.

18 38 Razmkhah, et al. 6 Concluding Remark In this paper, we have tackled the problem of reconstruction of missing order data. Several methods proposed and we applied them to reconstruct the missing order statistics based on the data set Y = {Y,..., Y r, Y s,..., Y n } from a two-parameter exponential distribution. Notice that by ) and 2), we have f Yl Yy l y) = e y r/σ e y l/σ ) l r e y l /σ e ys/σ) s l Bl r, s l) e yr/σ e ys/σ) s r e y l/σ σ. 39) The density function in 39) does not depend on the location parameter µ. The reconstructors were obtained in the case that σ is known. If the scale parameter is unknown, as mentioned in Balakrishnan et al. 2009), we can substitute a common estimator of σ, for example maximum likelihood estimator MLE), in the corresponding formulas. Let X,, X n be iid random variables with cdf F θ x) and pdf f θ x), where θ is an unknown parameter. Then, the likelihood function of θ on the basis of Y is Lθ) = n! s r )! i r,s f θ y i )[F θ y s ) F θ y r )] s r, 40) where r,s = {,..., r, s,..., n}. For the exponential distribution ), we consider three cases as follows: Case. 0 = r < s n left censored sample) As mentioned in Section, this case coincides with the left censored sample. Then, MLEµ) = Y s and hence using 40), MLEσ) = n s + n i=s+ W s,i. Case 2. r < s = n + right censored sample) This special case coincides with the Type II censored sample. Then MLEµ) = Y and hence using 40), MLEσ) = r { r W,i + n r + )W,r }. i=

19 Reconstruction of Order Statistics in Exponential Distribution 39 Case 3. r < s n. Using ), we have MLEµ) = Y and by 40) the MLE of σ is the solution of the following equation e W r,s/σ ) A n s r) + )σ ) = s r )W r,s, 4) where A = i r,s W,i + s r )W,r. 42) The solution Eq. 4) in terms of σ has no any closed form. Expanding the exponential function e x in a Taylor series, we obtain an approximation for the MLE of σ. By using the first two terms of the Taylor series, we find MLEσ) = { } W,i + s r )W,r. 43) n i r,s If more precision is needed in the approximation, then we use more terms of Taylor series expansion. Using the first j j > 2) terms of this series, the MLE of σ can be approximately found by solving the following polynomial equation j i=2 σ j i i! iaw i 2 r,s where A is defined in 42). Acknowledgments n s r) + )Wr,s i ) AWr,s j 2 + j )! = nσj, The authors thank the referees and the associate editor for their useful comments and constructive criticisms on the original version of this manuscript which led to this considerably improved version. Partial support from Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad is acknowledged. References Ahmadi, J., Jafari Jozani, M., Marchand, E., and Parsian, A. 2009), Prediction of k-records from a general class of distributions under balanced type loss functions. Metrika, 70, 9 33.

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